All atomic nuclei of elements with an odd atomic mass or an odd atomic number possess a nuclear magnetic moment. Nuclear magnetic resonance is a phenomenon exhibited by this select group of atomic nuclei (termed "NMR active" nuclei), and is based upon the interaction of the nucleus with an applied, external magnetic field. The magnetic properties of a nucleus are conveniently discussed in terms of two quantities: the gyromagnetic ratio (.gamma.); and the nuclear spin (I). When an NMR active nucleus is placed in a magnetic field, its nuclear magnetic energy levels are split into (2I+1) non-degenerate energy levels, which are separated from each other by an energy difference that is directly proportional to the strength of the applied magnetic field. This splitting is called the "Zeeman" splitting and the energy difference is equal to hH.sub.o /2.pi. where h is Plank's constant and H.sub.0 is the strength of the applied magnetic field. The frequency corresponding to the energy of the Zeeman splitting (.omega.=.gamma.H.sub.0) is called the "Larmor frequency" and is proportional to the field strength of the magnetic field. Typical NMR active nuclei include .sup.1 H (protons), .sup.13 C, .sup.19 F, and .sup.31 P. For these four nuclei I=1/2, and each nucleus has two nuclear magnetic energy levels.
When a bulk sample of material containing NMR active nuclei is placed within a magnetic field called the main static field, the nuclear spins distribute themselves amongst the nuclear magnetic energy levels in accordance with Boltzmann's statistics. This results in a population imbalance among the energy levels and a net nuclear magnetization. It is this net nuclear magnetization that is studied by NMR techniques.
At equilibrium, the net nuclear magnetization of the aforementioned bulk sample is aligned parallel to the external magnetic field and is static (by convention, the direction of the main static field is taken to be the z-axis). A second magnetic field perpendicular to the main static magnetic field and rotating at, or near, the Larmor frequency can be applied to induce a coherent motion of the net nuclear magnetization. Since, at conventional main static magnetic field strengths, the Larmor frequency is in the megahertz frequency range, this second magnetic field is called a "radio frequency" or RF field.
The effect of the RF field is to shift the nuclear magnetization direction so that it is no longer parallel to the main static field. This shift introduces a net coherent motion of the nuclear magnetization about the main static field direction called a "nutation". In order to conveniently deal with this nutation, a reference frame is used which rotates about the laboratory reference frame z-axis at the Larmor frequency and also has its z-axis parallel to the main static field direction. In this "rotating frame" the net nuclear magnetization, which is rotating in the stationary "laboratory" reference frame, is now static.
Consequently, the effect of the RF field is to rotate the now static nuclear magnetization direction at an angle with respect to the main static field direction (z-axis). The new magnetization direction can be broken into a component which is parallel to the main field direction (z-axis direction) and a component which lies in the plane transverse to the main magnetization (x,y plane). The RF field is typically applied in pulses of varying length and amplitude and, by convention, an RF pulse of sufficient amplitude and length to rotate the nuclear magnetization in the rotating frame through an angle of 90.degree., or .pi./2 radians, and entirely into the x,y plane is called a ".pi./2 pulse".
Because the net nuclear magnetization is rotating with respect to the laboratory frame, the component of the nuclear magnetization that is transverse to the main magnetic field or that lies in the x,y plane rotates about the external magnetic field at the Larmor frequency. This rotation can be detected with a receiver coil that is resonant at the Larmor frequency. The receiver coil is generally located so that it senses voltage changes along one axis (for example, the x-axis) where the rotating magnetization component appears as an oscillating voltage. Frequently, the "transmitter coil" employed for applying the RF field to the sample and the "receiver coil" employed for detecting the magnetization are one and the same coil.
Although the main static field is applied to the overall material sample, the nuclear magnetic moment in each nucleus within the sample actually experiences an external magnetic field that is changed from the main static field value due to a screening from the surrounding electron cloud. This screening results in a slight shift in the Larmor frequency for that nucleus (called the "chemical shift" since the size and symmetry of the shielding effect is dependent on the chemical composition of the sample).
In a typical NMR experiment, the sample is placed in the main static field and a .pi./2 pulse is applied to shift the net magnetization into the transverse plane (called transverse magnetization). After application of the pulse, the transverse magnetization, or "coherence", begins to precess about the x-axis, or evolve, due to the chemical shifts at a frequency which is proportional to the chemical shift field strength. In the rotating frame, the detector (which is stationary in the laboratory frame) appears to rotate at the Larmor frequency. Consequently, the detector senses an oscillation produced by an apparent magnetization rotation at a frequency which is proportional to the frequency difference between the Larmor frequency and the chemical shift frequency.
Thus, the detected signal oscillates at the frequency shift difference. In addition to precessing at the Larmor frequency, in the absence of the applied RF field energy, the nuclear magnetization also undergoes two spontaneous processes: (1) the precessions of various individual nuclear spins which generate the net nuclear magnetization become dephased with respect to each other so that the magnetization within the transverse plane loses phase coherence (so-called "spin-spin relaxation") with an associated relaxation time, T.sub.2, and (2) the individual nuclear spins return to their equilibrium population of the nuclear magnetic energy levels (so-called "spin-lattice relaxation") with an associated relaxation time, T.sub.1. The latter process causes the received signal to decay to zero. The decaying, oscillating signal is called a free induction decay (FID).
Many NMR experiments are designed such that the spin dynamics are uniform through the sample and, in these cases, the sample is placed in a uniform magnetic field. However, there are cases where it is advantageous to impose a spatial variation in the spin dynamics across the sample. Obvious examples of such cases include imaging experiments and diffusion experiments. The spin evolution may also be spatially modulated as a means of selecting a specific coherence pathway or multiple quantum spin state. Such a spatial variation may be introduced into an experiment by utilizing a magnetic field having a spatial variation (such as a linear gradient) to perform the experiment. Two magnetic fields are commonly employed in NMR experiments, B.sub.0 and B.sub.1 fields, which are fields that are oriented along the direction of the main static field and in the plane transverse to the main static field direction, respectively.
B.sub.1 -gradients are commonly generated by a special RF coil which produces a magnetic field that has at least one component with a direction lying in the plane perpendicular to the main static field direction. When B.sub.1 -gradient magnetic fields are used in NMR spectroscopy experiments, the gradient magnetic field is often used in conjunction with a homogeneous RF magnetic field which is also generated by an RF coil. The homogeneous field and the gradient field can be generated either by two different RF coils or by a single coil which can alternately be driven in a gradient mode and a homogeneous mode.
An example of the use of B.sub.1 -gradients in NMR spectroscopy has been described in detail in a paper entitled "The Selection of Coherence-Transfer Pathways by Inhomogeneous Z Pulses" by C. J. R. Counsell, M. H. Levitt and R. R. Ernst, Journal of Magnetic Resonance, v.64, pages 470-478, 1985. Another example of the use of B.sub.1 -gradients in NMR spectroscopy has been described in detail in another paper entitled "The Equivalent of the DQF-COSY Experiment, with One Transient Per t.sub.1 Value, by Use of B.sub.1 Gradients", by J. Brondeau, D. Boudot, P. Mutzenhardt and D. Canet, Journal of Magnetic Resonance, v. 100, pp. 611-618, 1992. B.sub.1 gradients have also been used for imaging by M. H. Werner and this approach is described in detail in a Ph.D. dissertation entitled, "NMR Imaging of Solids with Multiple-Pulse Line Narrowing and Radiofrequency Gradients" (M. H. Werner: Ph.D. Thesis, California Institute of Technology, Pasadena, Calif., 1993).
Since the spin dynamics are relative to the homogeneous RF field whose phase may itself vary over the sample, it is understood that it is the uniformity of the relative phase difference between the gradient and the homogeneous RF fields which is important. To simplify this discussion, however, the homogeneous RF field is discussed as though it generated an ideal uniform field.
An ideal field having a constant spatial field direction (or spatially-independent phase) can, in principle, be generated by means of an RF coil with proper geometry. However, practical requirements, such as RF efficiency, the frequency at which the coil is operated, and physical limitations on the size and placement of the gradient RF coil generally make it nearly impossible to generate an RF gradient magnetic field with a truly spatially independent phase.
For many applications involving B.sub.1 -gradients, the spatial variation of the magnetic field should ideally be only a variation in magnitude of the field, with the direction of the field, or the phase, remaining constant over space. A spatial dependence of the phase causes signal reduction and artifacts in experiments where the B.sub.1 -gradient field is used in combination with a homogeneous RF field.
Accordingly, it is an object of the present invention to provide a pulse sequence that converts an RF gradient field with a spatial variation in both amplitude and phase to an RF gradient field with a spatially dependent amplitude and a spatially independent phase.